How do I get an object to stretch between two animated objects?
I can suggest two options:


if you want to use it in 3d engines (with only bones animation), you can use two bone system:


One bone at one black thing and weight paint it (gradient) to the parallel part of the rubber mesh.
Second bone at the other black part
attach "track to" modifier to the first bone with target as second bone.

Make a "Shape Key" of the rubber with stretched pose, and move and rotate as required. You can also use both (1 &amp; 2) for easy animation and right mesh deformation.